The reduced $\operatorname{Sb}^d$ conjecture

Let $(X,H)$ be a smooth polarized variety over $\mathbb{C}$ and let $d\geq0$. Let $\mathfrak B_n^{>d}$ and $\pm\mathfrak B_n^{>d}$ denote the reduced-central-charge spaces satisfying the separation bound $\operatorname{sep}(\ell(\underline s,\underline t))>d$. Reduced $\operatorname{Sb}^d$ conjecture. There exists a family of reduced stability conditions $\operatorname{Sb}_H^{*>d}(X)$ with respect to the $H$-polarized lattice $\Lambda_H$ such that

$$\operatorname{Forg}:\operatorname{Sb}_H^{*>d}(X)\to(\Lambda_{\mathbb R})^*,\qquad \widetilde\sigma=(\mathcal A,B)\mapsto B$$

is a homeomorphism onto $\mathfrak B_n^{>d}$, the induced map

$$\operatorname{Forg}:\coprod_{n\in\mathbb Z}\operatorname{Sb}_H^{*>d}(X)[n]\to\pm\mathfrak B_n^{>d}$$

is a universal cover, tensoring by $\mathcal O_X(H)$ preserves the family, and for $\underline s<\underline t<\underline s[1]$ with $\operatorname{sep}(\ell(\underline s,\underline t))>d$ one has

$$\widetilde\sigma_{\underline s}\lesssim\widetilde\sigma_{\underline t}\lesssim\widetilde\sigma_{\underline s}[1].$$

This is the reduced counterpart of the $\operatorname{Stab}^d$ conjecture and extends the proposed reduced-stability picture beyond the cases motivating the construction; it remains open in the supplied text.